# Author: Raphael Vallat <raphaelvallat9@gmail.com>
import warnings
import numpy as np
import pandas as pd
import pandas_flavor as pf
from scipy.spatial.distance import pdist, squareform
from scipy.stats import pearsonr, spearmanr, kendalltau
from pingouin.config import options
from pingouin.power import power_corr
from pingouin.multicomp import multicomp
from pingouin.effsize import compute_esci
from pingouin.utils import remove_na, _perm_pval, _postprocess_dataframe
from pingouin.bayesian import bayesfactor_pearson
__all__ = ["corr", "partial_corr", "pcorr", "rcorr", "rm_corr", "distance_corr"]
def _correl_pvalue(r, n, k=0, alternative="two-sided"):
"""Compute the p-value of a correlation coefficient.
https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.pearsonr.html
https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#Using_the_exact_distribution
See also scipy.stats._ttest_finish
Parameters
----------
r : float
Correlation coefficient.
n : int
Sample size
k : int
Number of covariates for (semi)-partial correlation.
alternative : string
Tail of the test.
Returns
-------
pval : float
p-value.
Notes
-----
This uses the same approach as :py:func:`scipy.stats.pearsonr` to calculate
the p-value (i.e. using a beta distribution)
"""
from scipy.stats import t
assert alternative in [
"two-sided",
"greater",
"less",
], "Alternative must be one of 'two-sided' (default), 'greater' or 'less'."
# Method 1: using a student T distribution
dof = n - k - 2
tval = r * np.sqrt(dof / (1 - r**2))
if alternative == "less":
pval = t.cdf(tval, dof)
elif alternative == "greater":
pval = t.sf(tval, dof)
elif alternative == "two-sided":
pval = 2 * t.sf(np.abs(tval), dof)
# Method 2: beta distribution (similar to scipy.stats.pearsonr, faster)
# from scipy.special import btdtr
# ab = (n - k) / 2 - 1
# pval = 2 * btdtr(ab, ab, 0.5 * (1 - abs(np.float64(r))))
return pval
def skipped(x, y, corr_type="spearman"):
"""Skipped correlation (Rousselet and Pernet 2012).
Parameters
----------
x, y : array_like
First and second set of observations. x and y must be independent.
corr_type : str
Method used to compute the correlation after outlier removal. Can be
either 'spearman' (default) or 'pearson'.
Returns
-------
r : float
Skipped correlation coefficient.
pval : float
Two-tailed p-value.
outliers : array of bool
Indicate if value is an outlier or not
Notes
-----
The skipped correlation involves multivariate outlier detection using a
projection technique (Wilcox, 2004, 2005). First, a robust estimator of
multivariate location and scatter, for instance the minimum covariance
determinant estimator (MCD; Rousseeuw, 1984; Rousseeuw and van Driessen,
1999; Hubert et al., 2008) is computed. Second, data points are
orthogonally projected on lines joining each of the data point to the
location estimator. Third, outliers are detected using a robust technique.
Finally, Spearman correlations are computed on the remaining data points
and calculations are adjusted by taking into account the dependency among
the remaining data points.
Code inspired by Matlab code from Cyril Pernet and Guillaume
Rousselet [1]_.
Requires scikit-learn.
References
----------
.. [1] Pernet CR, Wilcox R, Rousselet GA. Robust Correlation Analyses:
False Positive and Power Validation Using a New Open Source Matlab
Toolbox. Frontiers in Psychology. 2012;3:606.
doi:10.3389/fpsyg.2012.00606.
"""
# Check that sklearn is installed
from pingouin.utils import _is_sklearn_installed
_is_sklearn_installed(raise_error=True)
from scipy.stats import chi2
from sklearn.covariance import MinCovDet
X = np.column_stack((x, y))
nrows, ncols = X.shape
gval = np.sqrt(chi2.ppf(0.975, 2))
# Compute center and distance to center
center = MinCovDet(random_state=42).fit(X).location_
# https://github.com/raphaelvallat/pingouin/issues/164
warnings.warn(
"The skipped correlation relies on the Minimum Covariance Determinant "
"algorithm, which gives slightly different results in Python "
"(scikit-learn) than in the original Matlab library (LIBRA). As such, "
"the skipped correlation may be different from the Matlab robust "
"correlation toolbox (see issue 164 on Pingouin's GitHub). "
"Make sure to double check your results or use another robust "
"correlation method."
)
B = X - center
bot = (B**2).sum(axis=1)
# Loop over rows
dis = np.zeros(shape=(nrows, nrows))
for i in np.arange(nrows):
if bot[i] != 0: # Avoid division by zero error
dis[i, :] = np.linalg.norm(B.dot(B[i, :, None]) * B[i, :] / bot[i], axis=1)
# Detect outliers
def idealf(x):
"""Compute the ideal fourths IQR (Wilcox 2012)."""
n = len(x)
j = int(np.floor(n / 4 + 5 / 12))
y = np.sort(x)
g = (n / 4) - j + (5 / 12)
low = (1 - g) * y[j - 1] + g * y[j]
k = n - j + 1
up = (1 - g) * y[k - 1] + g * y[k - 2]
return up - low
# One can either use the MAD or the IQR (see Wilcox 2012)
# MAD = mad(dis, axis=1)
iqr = np.apply_along_axis(idealf, 1, dis)
thresh = np.median(dis, axis=1) + gval * iqr
outliers = np.apply_along_axis(np.greater, 0, dis, thresh).any(axis=0)
# Compute correlation on remaining data
if corr_type == "spearman":
r, pval = spearmanr(X[~outliers, 0], X[~outliers, 1])
else:
r, pval = pearsonr(X[~outliers, 0], X[~outliers, 1])
return r, pval, outliers
def bsmahal(a, b, n_boot=200):
"""
Bootstraps Mahalanobis distances for Shepherd's pi correlation.
Parameters
----------
a : ndarray (shape=(n, 2))
Data
b : ndarray (shape=(n, 2))
Data
n_boot : int
Number of bootstrap samples to calculate.
Returns
-------
m : ndarray (shape=(n,))
Mahalanobis distance for each row in a, averaged across all the
bootstrap resamples.
"""
n, m = b.shape
MD = np.zeros((n, n_boot))
nr = np.arange(n)
xB = np.random.choice(nr, size=(n_boot, n), replace=True)
# Bootstrap the MD
for i in np.arange(n_boot):
s1 = b[xB[i, :], 0]
s2 = b[xB[i, :], 1]
X = np.column_stack((s1, s2))
mu = X.mean(0)
_, R = np.linalg.qr(X - mu)
sol = np.linalg.solve(R.T, (a - mu).T)
MD[:, i] = np.sum(sol**2, 0) * (n - 1)
# Average across all bootstraps
return MD.mean(1)
def shepherd(x, y, n_boot=200):
"""
Shepherd's Pi correlation, equivalent to Spearman's rho after outliers
removal.
Parameters
----------
x, y : array_like
First and second set of observations. x and y must be independent.
n_boot : int
Number of bootstrap samples to calculate.
Returns
-------
r : float
Pi correlation coefficient
pval : float
Two-tailed adjusted p-value.
outliers : array of bool
Indicate if value is an outlier or not
Notes
-----
It first bootstraps the Mahalanobis distances, removes all observations
with m >= 6 and finally calculates the correlation of the remaining data.
Pi is Spearman's Rho after outlier removal.
"""
X = np.column_stack((x, y))
# Bootstrapping on Mahalanobis distance
m = bsmahal(X, X, n_boot)
# Determine outliers
outliers = m >= 6
# Compute correlation
r, pval = spearmanr(x[~outliers], y[~outliers])
# (optional) double the p-value to achieve a nominal false alarm rate
# pval *= 2
# pval = 1 if pval > 1 else pval
return r, pval, outliers
def percbend(x, y, beta=0.2):
"""
Percentage bend correlation (Wilcox 1994).
Parameters
----------
x, y : array_like
First and second set of observations. x and y must be independent.
beta : float
Bending constant for omega (0 <= beta <= 0.5).
Returns
-------
r : float
Percentage bend correlation coefficient.
pval : float
Two-tailed p-value.
Notes
-----
Code inspired by Matlab code from Cyril Pernet and Guillaume Rousselet.
References
----------
.. [1] Wilcox, R.R., 1994. The percentage bend correlation coefficient.
Psychometrika 59, 601–616. https://doi.org/10.1007/BF02294395
.. [2] Pernet CR, Wilcox R, Rousselet GA. Robust Correlation Analyses:
False Positive and Power Validation Using a New Open Source Matlab
Toolbox. Frontiers in Psychology. 2012;3:606.
doi:10.3389/fpsyg.2012.00606.
"""
X = np.column_stack((x, y))
nx = X.shape[0]
M = np.tile(np.median(X, axis=0), nx).reshape(X.shape)
W = np.sort(np.abs(X - M), axis=0)
m = int((1 - beta) * nx)
omega = W[m - 1, :]
P = (X - M) / omega
P[np.isinf(P)] = 0
P[np.isnan(P)] = 0
# Loop over columns
a = np.zeros((2, nx))
for c in [0, 1]:
psi = P[:, c]
i1 = np.where(psi < -1)[0].size
i2 = np.where(psi > 1)[0].size
s = X[:, c].copy()
s[np.where(psi < -1)[0]] = 0
s[np.where(psi > 1)[0]] = 0
pbos = (np.sum(s) + omega[c] * (i2 - i1)) / (s.size - i1 - i2)
a[c] = (X[:, c] - pbos) / omega[c]
# Bend
a[a <= -1] = -1
a[a >= 1] = 1
# Correlation coefficient
a, b = a
r = (a * b).sum() / np.sqrt((a**2).sum() * (b**2).sum())
pval = _correl_pvalue(r, nx, k=0)
return r, pval
def bicor(x, y, c=9):
"""
Biweight midcorrelation.
Parameters
----------
x, y : array_like
First and second set of observations. x and y must be independent.
c : float
Tuning constant for the biweight estimator (default = 9.0).
Returns
-------
r : float
Correlation coefficient.
pval : float
Two-tailed p-value.
Notes
-----
This function will return (np.nan, np.nan) if mad(x) == 0 or mad(y) == 0.
References
----------
https://en.wikipedia.org/wiki/Biweight_midcorrelation
https://docs.astropy.org/en/stable/api/astropy.stats.biweight.biweight_midcovariance.html
Langfelder, P., & Horvath, S. (2012). Fast R Functions for Robust
Correlations and Hierarchical Clustering. Journal of Statistical Software,
46(11). https://www.ncbi.nlm.nih.gov/pubmed/23050260
"""
# Calculate median
nx = x.size
x_median = np.median(x)
y_median = np.median(y)
# Raw median absolute deviation
x_mad = np.median(np.abs(x - x_median))
y_mad = np.median(np.abs(y - y_median))
if x_mad == 0 or y_mad == 0:
# From Langfelder and Horvath 2012:
# "Strictly speaking, a call to bicor in R should return a missing
# value if mad(x) = 0 or mad(y) = 0." This avoids division by zero.
return np.nan, np.nan
# Calculate weights
u = (x - x_median) / (c * x_mad)
v = (y - y_median) / (c * y_mad)
w_x = (1 - u**2) ** 2 * ((1 - np.abs(u)) > 0)
w_y = (1 - v**2) ** 2 * ((1 - np.abs(v)) > 0)
# Normalize x and y by weights
x_norm = (x - x_median) * w_x
y_norm = (y - y_median) * w_y
denom = np.sqrt((x_norm**2).sum()) * np.sqrt((y_norm**2).sum())
# Correlation coefficient
r = (x_norm * y_norm).sum() / denom
pval = _correl_pvalue(r, nx, k=0)
return r, pval
[docs]
def corr(x, y, alternative="two-sided", method="pearson", **kwargs):
"""(Robust) correlation between two variables.
Parameters
----------
x, y : array_like
First and second set of observations. ``x`` and ``y`` must be
independent.
alternative : string
Defines the alternative hypothesis, or tail of the correlation. Must be one of
"two-sided" (default), "greater" or "less". Both "greater" and "less" return a one-sided
p-value. "greater" tests against the alternative hypothesis that the correlation is
positive (greater than zero), "less" tests against the hypothesis that the correlation is
negative.
method : string
Correlation type:
* ``'pearson'``: Pearson :math:`r` product-moment correlation
* ``'spearman'``: Spearman :math:`\\rho` rank-order correlation
* ``'kendall'``: Kendall's :math:`\\tau_B` correlation (for ordinal data)
* ``'bicor'``: Biweight midcorrelation (robust)
* ``'percbend'``: Percentage bend correlation (robust)
* ``'shepherd'``: Shepherd's pi correlation (robust)
* ``'skipped'``: Skipped correlation (robust)
**kwargs : optional
Optional argument(s) passed to the lower-level correlation functions.
Returns
-------
stats : :py:class:`pandas.DataFrame`
* ``'n'``: Sample size (after removal of missing values)
* ``'outliers'``: number of outliers, only if a robust method was used
* ``'r'``: Correlation coefficient
* ``'CI95%'``: 95% parametric confidence intervals around :math:`r`
* ``'p-val'``: p-value
* ``'BF10'``: Bayes Factor of the alternative hypothesis (only for Pearson correlation)
* ``'power'``: achieved power of the test with an alpha of 0.05.
See also
--------
pairwise_corr : Pairwise correlation between columns of a pandas DataFrame
partial_corr : Partial correlation
rm_corr : Repeated measures correlation
Notes
-----
The `Pearson correlation coefficient
<https://en.wikipedia.org/wiki/Pearson_correlation_coefficient>`_
measures the linear relationship between two datasets. Strictly speaking,
Pearson's correlation requires that each dataset be normally distributed.
Correlations of -1 or +1 imply a perfect negative and positive linear
relationship, respectively, with 0 indicating the absence of association.
.. math::
r_{xy} = \\frac{\\sum_i(x_i - \\bar{x})(y_i - \\bar{y})}
{\\sqrt{\\sum_i(x_i - \\bar{x})^2} \\sqrt{\\sum_i(y_i - \\bar{y})^2}}
= \\frac{\\text{cov}(x, y)}{\\sigma_x \\sigma_y}
where :math:`\\text{cov}` is the sample covariance and :math:`\\sigma`
is the sample standard deviation.
If ``method='pearson'``, The Bayes Factor is calculated using the
:py:func:`pingouin.bayesfactor_pearson` function.
The `Spearman correlation coefficient
<https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient>`_
is a non-parametric measure of the monotonicity of the relationship between
two datasets. Unlike the Pearson correlation, the Spearman correlation does
not assume that both datasets are normally distributed. Correlations of -1
or +1 imply an exact negative and positive monotonic relationship,
respectively. Mathematically, the Spearman correlation coefficient is
defined as the Pearson correlation coefficient between the
`rank variables <https://en.wikipedia.org/wiki/Ranking>`_.
The `Kendall correlation coefficient
<https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient>`_
is a measure of the correspondence between two rankings. Values also range
from -1 (perfect disagreement) to 1 (perfect agreement), with 0 indicating
the absence of association. Consistent with
:py:func:`scipy.stats.kendalltau`, Pingouin returns the Tau-b coefficient,
which adjusts for ties:
.. math:: \\tau_B = \\frac{(P - Q)}{\\sqrt{(P + Q + T) (P + Q + U)}}
where :math:`P` is the number of concordant pairs, :math:`Q` the number of
discordand pairs, :math:`T` the number of ties in x, and :math:`U`
the number of ties in y.
The `biweight midcorrelation
<https://en.wikipedia.org/wiki/Biweight_midcorrelation>`_ and
percentage bend correlation [1]_ are both robust methods that
protects against *univariate* outliers by down-weighting observations that
deviate too much from the median.
The Shepherd pi [2]_ correlation and skipped [3]_, [4]_ correlation are
both robust methods that returns the Spearman correlation coefficient after
removing *bivariate* outliers. Briefly, the Shepherd pi uses a
bootstrapping of the Mahalanobis distance to identify outliers, while the
skipped correlation is based on the minimum covariance determinant
(which requires scikit-learn). Note that these two methods are
significantly slower than the previous ones.
The confidence intervals for the correlation coefficient are estimated
using the Fisher transformation.
.. important:: Rows with missing values (NaN) are automatically removed.
References
----------
.. [1] Wilcox, R.R., 1994. The percentage bend correlation coefficient.
Psychometrika 59, 601–616. https://doi.org/10.1007/BF02294395
.. [2] Schwarzkopf, D.S., De Haas, B., Rees, G., 2012. Better ways to
improve standards in brain-behavior correlation analysis. Front.
Hum. Neurosci. 6, 200. https://doi.org/10.3389/fnhum.2012.00200
.. [3] Rousselet, G.A., Pernet, C.R., 2012. Improving standards in
brain-behavior correlation analyses. Front. Hum. Neurosci. 6, 119.
https://doi.org/10.3389/fnhum.2012.00119
.. [4] Pernet, C.R., Wilcox, R., Rousselet, G.A., 2012. Robust correlation
analyses: false positive and power validation using a new open
source matlab toolbox. Front. Psychol. 3, 606.
https://doi.org/10.3389/fpsyg.2012.00606
Examples
--------
1. Pearson correlation
>>> import numpy as np
>>> import pingouin as pg
>>> # Generate random correlated samples
>>> np.random.seed(123)
>>> mean, cov = [4, 6], [(1, .5), (.5, 1)]
>>> x, y = np.random.multivariate_normal(mean, cov, 30).T
>>> # Compute Pearson correlation
>>> pg.corr(x, y).round(3)
n r CI95% p-val BF10 power
pearson 30 0.491 [0.16, 0.72] 0.006 8.55 0.809
2. Pearson correlation with two outliers
>>> x[3], y[5] = 12, -8
>>> pg.corr(x, y).round(3)
n r CI95% p-val BF10 power
pearson 30 0.147 [-0.23, 0.48] 0.439 0.302 0.121
3. Spearman correlation (robust to outliers)
>>> pg.corr(x, y, method="spearman").round(3)
n r CI95% p-val power
spearman 30 0.401 [0.05, 0.67] 0.028 0.61
4. Biweight midcorrelation (robust)
>>> pg.corr(x, y, method="bicor").round(3)
n r CI95% p-val power
bicor 30 0.393 [0.04, 0.66] 0.031 0.592
5. Percentage bend correlation (robust)
>>> pg.corr(x, y, method='percbend').round(3)
n r CI95% p-val power
percbend 30 0.389 [0.03, 0.66] 0.034 0.581
6. Shepherd's pi correlation (robust)
>>> pg.corr(x, y, method='shepherd').round(3)
n outliers r CI95% p-val power
shepherd 30 2 0.437 [0.08, 0.7] 0.02 0.662
7. Skipped spearman correlation (robust)
>>> pg.corr(x, y, method='skipped').round(3)
n outliers r CI95% p-val power
skipped 30 2 0.437 [0.08, 0.7] 0.02 0.662
8. One-tailed Pearson correlation
>>> pg.corr(x, y, alternative="greater", method='pearson').round(3)
n r CI95% p-val BF10 power
pearson 30 0.147 [-0.17, 1.0] 0.22 0.467 0.194
>>> pg.corr(x, y, alternative="less", method='pearson').round(3)
n r CI95% p-val BF10 power
pearson 30 0.147 [-1.0, 0.43] 0.78 0.137 0.008
9. Perfect correlation
>>> pg.corr(x, -x).round(3)
n r CI95% p-val BF10 power
pearson 30 -1.0 [-1.0, -1.0] 0.0 inf 1
10. Using columns of a pandas dataframe
>>> import pandas as pd
>>> data = pd.DataFrame({'x': x, 'y': y})
>>> pg.corr(data['x'], data['y']).round(3)
n r CI95% p-val BF10 power
pearson 30 0.147 [-0.23, 0.48] 0.439 0.302 0.121
"""
# Safety check
x = np.asarray(x)
y = np.asarray(y)
assert x.ndim == y.ndim == 1, "x and y must be 1D array."
assert x.size == y.size, "x and y must have the same length."
assert alternative in [
"two-sided",
"greater",
"less",
], "Alternative must be one of 'two-sided' (default), 'greater' or 'less'."
if "tail" in kwargs:
raise ValueError(
"Since Pingouin 0.4.0, the 'tail' argument has been renamed to 'alternative'."
)
# Remove rows with missing values
x, y = remove_na(x, y, paired=True)
n = x.size
# Compute correlation coefficient and two-sided p-value
if method == "pearson":
r, pval = pearsonr(x, y)
elif method == "spearman":
r, pval = spearmanr(x, y, **kwargs)
elif method == "kendall":
r, pval = kendalltau(x, y, **kwargs)
elif method == "bicor":
r, pval = bicor(x, y, **kwargs)
elif method == "percbend":
r, pval = percbend(x, y, **kwargs)
elif method == "shepherd":
r, pval, outliers = shepherd(x, y, **kwargs)
elif method == "skipped":
r, pval, outliers = skipped(x, y, **kwargs)
else:
raise ValueError(f'Method "{method}" not recognized.')
if np.isnan(r):
# Correlation failed -- new in version v0.3.4, instead of raising an
# error we just return a dataframe full of NaN (except sample size).
# This avoid sudden stop in pingouin.pairwise_corr.
return pd.DataFrame(
{
"n": n,
"r": np.nan,
"CI95%": np.nan,
"p-val": np.nan,
"BF10": np.nan,
"power": np.nan,
},
index=[method],
)
# Sample size after outlier removal
n_outliers = sum(outliers) if "outliers" in locals() else 0
n_clean = n - n_outliers
# Rounding errors caused an r value marginally beyond 1
if abs(r) > 1 and np.isclose(abs(r), 1):
r = np.clip(r, -1, 1)
# Compute the parametric 95% confidence interval and power
if abs(r) == 1:
ci = [r, r]
pr = 1
else:
ci = compute_esci(
stat=r, nx=n_clean, ny=n_clean, eftype="r", decimals=6, alternative=alternative
)
pr = power_corr(r=r, n=n_clean, power=None, alpha=0.05, alternative=alternative)
# Recompute p-value if tail is one-sided
if alternative != "two-sided":
pval = _correl_pvalue(r, n_clean, k=0, alternative=alternative)
# Create dictionnary
stats = {"n": n, "r": r, "CI95%": [ci], "p-val": pval, "power": pr}
if method in ["shepherd", "skipped"]:
stats["outliers"] = n_outliers
# Compute the BF10 for Pearson correlation only
if method == "pearson":
stats["BF10"] = bayesfactor_pearson(r, n_clean, alternative=alternative)
# Convert to DataFrame
stats = pd.DataFrame(stats, index=[method])
# Define order
col_keep = ["n", "outliers", "r", "CI95%", "p-val", "BF10", "power"]
col_order = [k for k in col_keep if k in stats.keys().tolist()]
return _postprocess_dataframe(stats)[col_order]
[docs]
@pf.register_dataframe_method
def partial_corr(
data=None,
x=None,
y=None,
covar=None,
x_covar=None,
y_covar=None,
alternative="two-sided",
method="pearson",
):
"""Partial and semi-partial correlation.
Parameters
----------
data : :py:class:`pandas.DataFrame`
Pandas Dataframe. Note that this function can also directly be used
as a :py:class:`pandas.DataFrame` method, in which case this argument
is no longer needed.
x, y : string
x and y. Must be names of columns in ``data``.
covar : string or list
Covariate(s). Must be a names of columns in ``data``. Use a list if
there are two or more covariates.
x_covar : string or list
Covariate(s) for the ``x`` variable. This is used to compute
semi-partial correlation (i.e. the effect of ``x_covar`` is removed
from ``x`` but not from ``y``). Only one of ``covar``, ``x_covar`` and
``y_covar`` can be specified.
y_covar : string or list
Covariate(s) for the ``y`` variable. This is used to compute
semi-partial correlation (i.e. the effect of ``y_covar`` is removed
from ``y`` but not from ``x``). Only one of ``covar``, ``x_covar`` and
``y_covar`` can be specified.
alternative : string
Defines the alternative hypothesis, or tail of the partial correlation. Must be one of
"two-sided" (default), "greater" or "less". Both "greater" and "less" return a one-sided
p-value. "greater" tests against the alternative hypothesis that the partial correlation is
positive (greater than zero), "less" tests against the hypothesis that the partial
correlation is negative.
method : string
Correlation type:
* ``'pearson'``: Pearson :math:`r` product-moment correlation
* ``'spearman'``: Spearman :math:`\\rho` rank-order correlation
Returns
-------
stats : :py:class:`pandas.DataFrame`
* ``'n'``: Sample size (after removal of missing values)
* ``'r'``: Partial correlation coefficient
* ``'CI95'``: 95% parametric confidence intervals around :math:`r`
* ``'p-val'``: p-value
See also
--------
corr, pcorr, pairwise_corr, rm_corr
Notes
-----
Partial correlation [1]_ measures the degree of association between ``x``
and ``y``, after removing the effect of one or more controlling variables
(``covar``, or :math:`Z`). Practically, this is achieved by calculating the
correlation coefficient between the residuals of two linear regressions:
.. math:: x \\sim Z, y \\sim Z
Like the correlation coefficient, the partial correlation
coefficient takes on a value in the range from –1 to 1, where 1 indicates a
perfect positive association.
The semipartial correlation is similar to the partial correlation,
with the exception that the set of controlling variables is only
removed for either ``x`` or ``y``, but not both.
Pingouin uses the method described in [2]_ to calculate the (semi)partial
correlation coefficients and associated p-values. This method is based on
the inverse covariance matrix and is significantly faster than the
traditional regression-based method. Results have been tested against the
`ppcor <https://cran.r-project.org/web/packages/ppcor/index.html>`_
R package.
.. important:: Rows with missing values are automatically removed from
data.
References
----------
.. [1] https://en.wikipedia.org/wiki/Partial_correlation
.. [2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4681537/
Examples
--------
1. Partial correlation with one covariate
>>> import pingouin as pg
>>> df = pg.read_dataset('partial_corr')
>>> pg.partial_corr(data=df, x='x', y='y', covar='cv1').round(3)
n r CI95% p-val
pearson 30 0.568 [0.25, 0.77] 0.001
2. Spearman partial correlation with several covariates
>>> # Partial correlation of x and y controlling for cv1, cv2 and cv3
>>> pg.partial_corr(data=df, x='x', y='y', covar=['cv1', 'cv2', 'cv3'],
... method='spearman').round(3)
n r CI95% p-val
spearman 30 0.521 [0.18, 0.75] 0.005
3. Same but one-sided test
>>> pg.partial_corr(data=df, x='x', y='y', covar=['cv1', 'cv2', 'cv3'],
... alternative="greater", method='spearman').round(3)
n r CI95% p-val
spearman 30 0.521 [0.24, 1.0] 0.003
>>> pg.partial_corr(data=df, x='x', y='y', covar=['cv1', 'cv2', 'cv3'],
... alternative="less", method='spearman').round(3)
n r CI95% p-val
spearman 30 0.521 [-1.0, 0.72] 0.997
4. As a pandas method
>>> df.partial_corr(x='x', y='y', covar=['cv1'], method='spearman').round(3)
n r CI95% p-val
spearman 30 0.578 [0.27, 0.78] 0.001
5. Partial correlation matrix (returns only the correlation coefficients)
>>> df.pcorr().round(3)
x y cv1 cv2 cv3
x 1.000 0.493 -0.095 0.130 -0.385
y 0.493 1.000 -0.007 0.104 -0.002
cv1 -0.095 -0.007 1.000 -0.241 -0.470
cv2 0.130 0.104 -0.241 1.000 -0.118
cv3 -0.385 -0.002 -0.470 -0.118 1.000
6. Semi-partial correlation on x
>>> pg.partial_corr(data=df, x='x', y='y', x_covar=['cv1', 'cv2', 'cv3']).round(3)
n r CI95% p-val
pearson 30 0.463 [0.1, 0.72] 0.015
"""
from pingouin.utils import _flatten_list
# Safety check
assert alternative in [
"two-sided",
"greater",
"less",
], "Alternative must be one of 'two-sided' (default), 'greater' or 'less'."
assert method in [
"pearson",
"spearman",
], 'only "pearson" and "spearman" are supported for partial correlation.'
assert isinstance(data, pd.DataFrame), "data must be a pandas DataFrame."
assert data.shape[0] > 2, "Data must have at least 3 samples."
if covar is not None and (x_covar is not None or y_covar is not None):
raise ValueError("Cannot specify both covar and {x,y}_covar.")
if x_covar is not None and y_covar is not None:
raise ValueError("Cannot specify both x_covar and y_covar.")
assert x != y, "x and y must be independent"
if isinstance(covar, list):
assert x not in covar, "x and covar must be independent"
assert y not in covar, "y and covar must be independent"
else:
assert x != covar, "x and covar must be independent"
assert y != covar, "y and covar must be independent"
# Check that columns exist
col = _flatten_list([x, y, covar, x_covar, y_covar])
assert all([c in data for c in col]), "columns are not in dataframe."
# Check that columns are numeric
assert all([data[c].dtype.kind in "bfiu" for c in col])
# Drop rows with NaN
data = data[col].dropna()
n = data.shape[0] # Number of samples
k = data.shape[1] - 2 # Number of covariates
assert n > 2, "Data must have at least 3 non-NAN samples."
# Calculate the partial corrrelation matrix - similar to pingouin.pcorr()
if method == "spearman":
# Convert the data to rank, similar to R cov()
V = data.rank(na_option="keep").cov(numeric_only=True)
else:
V = data.cov(numeric_only=True)
Vi = np.linalg.pinv(V, hermitian=True) # Inverse covariance matrix
Vi_diag = Vi.diagonal()
D = np.diag(np.sqrt(1 / Vi_diag))
pcor = -1 * (D @ Vi @ D) # Partial correlation matrix
if covar is not None:
r = pcor[0, 1]
else:
# Semi-partial correlation matrix
with np.errstate(divide="ignore"):
spcor = (
pcor
/ np.sqrt(np.diag(V))[..., None]
/ np.sqrt(np.abs(Vi_diag - Vi**2 / Vi_diag[..., None])).T
)
if y_covar is not None:
r = spcor[0, 1] # y_covar is removed from y
else:
r = spcor[1, 0] # x_covar is removed from x
if np.isnan(r):
# Correlation failed. Return NaN. When would this happen?
return pd.DataFrame({"n": n, "r": np.nan, "CI95%": np.nan, "p-val": np.nan}, index=[method])
# Compute the two-sided p-value and confidence intervals
# https://online.stat.psu.edu/stat505/lesson/6/6.3
pval = _correl_pvalue(r, n, k, alternative)
ci = compute_esci(
stat=r, nx=(n - k), ny=(n - k), eftype="r", decimals=6, alternative=alternative
)
# Create dictionnary
stats = {
"n": n,
"r": r,
"CI95%": [ci],
"p-val": pval,
}
# Convert to DataFrame
stats = pd.DataFrame(stats, index=[method])
# Define order
col_keep = ["n", "r", "CI95%", "p-val"]
col_order = [k for k in col_keep if k in stats.keys().tolist()]
return _postprocess_dataframe(stats)[col_order]
[docs]
@pf.register_dataframe_method
def pcorr(self):
"""Partial correlation matrix (:py:class:`pandas.DataFrame` method).
Returns
-------
pcormat : :py:class:`pandas.DataFrame`
Partial correlation matrix.
Notes
-----
This function calculates the pairwise partial correlations for each pair of
variables in a :py:class:`pandas.DataFrame` given all the others. It has
the same behavior as the pcor function in the
`ppcor <https://cran.r-project.org/web/packages/ppcor/index.html>`_
R package.
Note that this function only returns the raw Pearson correlation
coefficient. If you want to calculate the test statistic and p-values, or
use more robust estimates of the correlation coefficient, please refer to
the :py:func:`pingouin.pairwise_corr` or :py:func:`pingouin.partial_corr`
functions.
Examples
--------
>>> import pingouin as pg
>>> data = pg.read_dataset('mediation')
>>> data.pcorr().round(3)
X M Y Mbin Ybin W1 W2
X 1.000 0.359 0.074 -0.019 -0.147 -0.148 -0.067
M 0.359 1.000 0.555 -0.024 -0.112 -0.138 -0.176
Y 0.074 0.555 1.000 -0.001 0.169 0.101 0.108
Mbin -0.019 -0.024 -0.001 1.000 -0.080 -0.032 -0.040
Ybin -0.147 -0.112 0.169 -0.080 1.000 -0.000 -0.140
W1 -0.148 -0.138 0.101 -0.032 -0.000 1.000 -0.394
W2 -0.067 -0.176 0.108 -0.040 -0.140 -0.394 1.000
On a subset of columns
>>> data[['X', 'Y', 'M']].pcorr()
X Y M
X 1.000000 0.036649 0.412804
Y 0.036649 1.000000 0.540140
M 0.412804 0.540140 1.000000
"""
V = self.cov(numeric_only=True) # Covariance matrix
Vi = np.linalg.pinv(V, hermitian=True) # Inverse covariance matrix
D = np.diag(np.sqrt(1 / np.diag(Vi)))
pcor = -1 * (D @ Vi @ D) # Partial correlation matrix
pcor[np.diag_indices_from(pcor)] = 1
return pd.DataFrame(pcor, index=V.index, columns=V.columns)
[docs]
@pf.register_dataframe_method
def rcorr(
self,
method="pearson",
upper="pval",
decimals=3,
padjust=None,
stars=True,
pval_stars={0.001: "***", 0.01: "**", 0.05: "*"},
):
"""
Correlation matrix of a dataframe with p-values and/or sample size on the
upper triangle (:py:class:`pandas.DataFrame` method).
This method is a faster, but less exhaustive, matrix-version of the
:py:func:`pingouin.pairwise_corr` function. It is based on the
:py:func:`pandas.DataFrame.corr` method. Missing values are automatically
removed from each pairwise correlation.
Parameters
----------
self : :py:class:`pandas.DataFrame`
Input dataframe.
method : str
Correlation method. Can be either 'pearson' or 'spearman'.
upper : str
If 'pval', the upper triangle of the output correlation matrix shows
the p-values. If 'n', the upper triangle is the sample size used in
each pairwise correlation.
decimals : int
Number of decimals to display in the output correlation matrix.
padjust : string or None
Method used for testing and adjustment of pvalues.
* ``'none'``: no correction
* ``'bonf'``: one-step Bonferroni correction
* ``'sidak'``: one-step Sidak correction
* ``'holm'``: step-down method using Bonferroni adjustments
* ``'fdr_bh'``: Benjamini/Hochberg FDR correction
* ``'fdr_by'``: Benjamini/Yekutieli FDR correction
stars : boolean
If True, only significant p-values are displayed as stars using the
pre-defined thresholds of ``pval_stars``. If False, all the raw
p-values are displayed.
pval_stars : dict
Significance thresholds. Default is 3 stars for p-values < 0.001,
2 stars for p-values < 0.01 and 1 star for p-values < 0.05.
Returns
-------
rcorr : :py:class:`pandas.DataFrame`
Correlation matrix, of type str.
Examples
--------
>>> import numpy as np
>>> import pandas as pd
>>> import pingouin as pg
>>> # Load an example dataset of personality dimensions
>>> df = pg.read_dataset('pairwise_corr').iloc[:, 1:]
>>> # Add some missing values
>>> df.iloc[[2, 5, 20], 2] = np.nan
>>> df.iloc[[1, 4, 10], 3] = np.nan
>>> df.head().round(2)
Neuroticism Extraversion Openness Agreeableness Conscientiousness
0 2.48 4.21 3.94 3.96 3.46
1 2.60 3.19 3.96 NaN 3.23
2 2.81 2.90 NaN 2.75 3.50
3 2.90 3.56 3.52 3.17 2.79
4 3.02 3.33 4.02 NaN 2.85
>>> # Correlation matrix on the four first columns
>>> df.iloc[:, 0:4].rcorr()
Neuroticism Extraversion Openness Agreeableness
Neuroticism - *** **
Extraversion -0.35 - ***
Openness -0.01 0.265 - ***
Agreeableness -0.134 0.054 0.161 -
>>> # Spearman correlation and Holm adjustement for multiple comparisons
>>> df.iloc[:, 0:4].rcorr(method='spearman', padjust='holm')
Neuroticism Extraversion Openness Agreeableness
Neuroticism - *** **
Extraversion -0.325 - ***
Openness -0.027 0.24 - ***
Agreeableness -0.15 0.06 0.173 -
>>> # Compare with the pg.pairwise_corr function
>>> pairwise = df.iloc[:, 0:4].pairwise_corr(method='spearman',
... padjust='holm')
>>> pairwise[['X', 'Y', 'r', 'p-corr']].round(3) # Do not show all columns
X Y r p-corr
0 Neuroticism Extraversion -0.325 0.000
1 Neuroticism Openness -0.027 0.543
2 Neuroticism Agreeableness -0.150 0.002
3 Extraversion Openness 0.240 0.000
4 Extraversion Agreeableness 0.060 0.358
5 Openness Agreeableness 0.173 0.000
>>> # Display the raw p-values with four decimals
>>> df.iloc[:, [0, 1, 3]].rcorr(stars=False, decimals=4)
Neuroticism Extraversion Agreeableness
Neuroticism - 0.0000 0.0028
Extraversion -0.3501 - 0.2305
Agreeableness -0.134 0.0539 -
>>> # With the sample size on the upper triangle instead of the p-values
>>> df.iloc[:, [0, 1, 2]].rcorr(upper='n')
Neuroticism Extraversion Openness
Neuroticism - 500 497
Extraversion -0.35 - 497
Openness -0.01 0.265 -
"""
from numpy import triu_indices_from as tif
from numpy import format_float_positional as ffp
from scipy.stats import pearsonr, spearmanr
# Safety check
assert isinstance(pval_stars, dict), "pval_stars must be a dictionnary."
assert isinstance(decimals, int), "decimals must be an int."
assert method in ["pearson", "spearman"], "Method is not recognized."
assert upper in ["pval", "n"], "upper must be either `pval` or `n`."
mat = self.corr(method=method, numeric_only=True).round(decimals)
if upper == "n":
mat_upper = self.corr(method=lambda x, y: len(x), numeric_only=True).astype(int)
else:
if method == "pearson":
mat_upper = self.corr(method=lambda x, y: pearsonr(x, y)[1], numeric_only=True)
else:
# Method = 'spearman'
mat_upper = self.corr(method=lambda x, y: spearmanr(x, y)[1], numeric_only=True)
if padjust is not None:
pvals = mat_upper.to_numpy()[tif(mat, k=1)]
mat_upper.to_numpy()[tif(mat, k=1)] = multicomp(pvals, alpha=0.05, method=padjust)[1]
# Convert r to text
mat = mat.astype(str)
# Inplace modification of the diagonal
np.fill_diagonal(mat.to_numpy(), "-")
if upper == "pval":
def replace_pval(x):
for key, value in pval_stars.items():
if x < key:
return value
return ""
if stars:
# Replace p-values by stars
mat_upper = mat_upper.map(replace_pval)
else:
mat_upper = mat_upper.map(lambda x: ffp(x, precision=decimals))
# Replace upper triangle by p-values or n
mat.to_numpy()[tif(mat, k=1)] = mat_upper.to_numpy()[tif(mat, k=1)]
return mat
[docs]
def rm_corr(data=None, x=None, y=None, subject=None):
"""Repeated measures correlation.
Parameters
----------
data : :py:class:`pandas.DataFrame`
Dataframe.
x, y : string
Name of columns in ``data`` containing the two dependent variables.
subject : string
Name of column in ``data`` containing the subject indicator.
Returns
-------
stats : :py:class:`pandas.DataFrame`
* ``'r'``: Repeated measures correlation coefficient
* ``'dof'``: Degrees of freedom
* ``'pval'``: p-value
* ``'CI95'``: 95% parametric confidence intervals
* ``'power'``: achieved power of the test (= 1 - type II error).
See also
--------
plot_rm_corr
Notes
-----
Repeated measures correlation (rmcorr) is a statistical technique for determining the common
within-individual association for paired measures assessed on two or more occasions for
multiple individuals.
From `Bakdash and Marusich (2017)
<https://doi.org/10.3389/fpsyg.2017.00456>`_:
*Rmcorr accounts for non-independence among observations using analysis
of covariance (ANCOVA) to statistically adjust for inter-individual
variability. By removing measured variance between-participants,
rmcorr provides the best linear fit for each participant using parallel
regression lines (the same slope) with varying intercepts.
Like a Pearson correlation coefficient, the rmcorr coefficient
is bounded by − 1 to 1 and represents the strength of the linear
association between two variables.*
Results have been tested against the `rmcorr <https://github.com/cran/rmcorr>`_ R package.
Missing values are automatically removed from the dataframe (listwise deletion).
Examples
--------
>>> import pingouin as pg
>>> df = pg.read_dataset('rm_corr')
>>> pg.rm_corr(data=df, x='pH', y='PacO2', subject='Subject')
r dof pval CI95% power
rm_corr -0.50677 38 0.000847 [-0.71, -0.23] 0.929579
Now plot using the :py:func:`pingouin.plot_rm_corr` function:
.. plot::
>>> import pingouin as pg
>>> df = pg.read_dataset('rm_corr')
>>> g = pg.plot_rm_corr(data=df, x='pH', y='PacO2', subject='Subject')
"""
from pingouin import ancova, power_corr
# Safety checks
assert isinstance(data, pd.DataFrame), "Data must be a DataFrame"
assert x in data.columns, "The %s column is not in data." % x
assert y in data.columns, "The %s column is not in data." % y
assert data[x].dtype.kind in "bfiu", "%s must be numeric." % x
assert data[y].dtype.kind in "bfiu", "%s must be numeric." % y
assert subject in data.columns, "The %s column is not in data." % subject
if data[subject].nunique() < 3:
raise ValueError("rm_corr requires at least 3 unique subjects.")
# Remove missing values
data = data[[x, y, subject]].dropna(axis=0)
# Using PINGOUIN
# For max precision, make sure rounding is disabled
old_options = options.copy()
options["round"] = None
aov = ancova(dv=y, covar=x, between=subject, data=data)
options.update(old_options) # restore options
bw = aov.bw_ # Beta within parameter
sign = np.sign(bw)
dof = int(aov.at[2, "DF"])
n = dof + 2
ssfactor = aov.at[1, "SS"]
sserror = aov.at[2, "SS"]
rm = sign * np.sqrt(ssfactor / (ssfactor + sserror))
pval = aov.at[1, "p-unc"]
ci = compute_esci(stat=rm, nx=n, eftype="pearson").tolist()
pwr = power_corr(r=rm, n=n, alternative="two-sided")
# Convert to Dataframe
stats = pd.DataFrame(
{"r": rm, "dof": int(dof), "pval": pval, "CI95%": [ci], "power": pwr}, index=["rm_corr"]
)
return _postprocess_dataframe(stats)
def _dcorr(y, n2, A, dcov2_xx):
"""Helper function for distance correlation bootstrapping."""
# Pairwise Euclidean distances
b = squareform(pdist(y, metric="euclidean"))
# Double centering
B = b - b.mean(axis=0)[None, :] - b.mean(axis=1)[:, None] + b.mean()
# Compute squared distance covariances
dcov2_yy = np.vdot(B, B) / n2
dcov2_xy = np.vdot(A, B) / n2
return np.sqrt(dcov2_xy) / np.sqrt(np.sqrt(dcov2_xx) * np.sqrt(dcov2_yy))
[docs]
def distance_corr(x, y, alternative="greater", n_boot=1000, seed=None):
"""Distance correlation between two arrays.
Statistical significance (p-value) is evaluated with a permutation test.
Parameters
----------
x, y : array_like
1D or 2D input arrays, shape (n_samples, n_features).
``x`` and ``y`` must have the same number of samples and must not
contain missing values.
alternative : str
Alternative of the test. Can be either "two-sided", "greater" (default) or "less".
To be consistent with the original R implementation, the default is to calculate the
one-sided "greater" p-value.
n_boot : int or None
Number of bootstrap to perform. If None, no bootstrapping is performed and the function
only returns the distance correlation (no p-value). Default is 1000 (thus giving a
precision of 0.001).
seed : int or None
Random state seed.
Returns
-------
dcor : float
Sample distance correlation (range from 0 to 1).
pval : float
P-value.
Notes
-----
From Wikipedia:
*Distance correlation is a measure of dependence between two paired
random vectors of arbitrary, not necessarily equal, dimension. The
distance correlation coefficient is zero if and only if the random
vectors are independent. Thus, distance correlation measures both
linear and nonlinear association between two random variables or
random vectors. This is in contrast to Pearson's correlation, which can
only detect linear association between two random variables.*
The distance correlation of two random variables is obtained by
dividing their distance covariance by the product of their distance
standard deviations:
.. math::
\\text{dCor}(X, Y) = \\frac{\\text{dCov}(X, Y)}
{\\sqrt{\\text{dVar}(X) \\cdot \\text{dVar}(Y)}}
where :math:`\\text{dCov}(X, Y)` is the square root of the arithmetic
average of the product of the double-centered pairwise Euclidean distance
matrices.
Note that by contrast to Pearson's correlation, the distance correlation
cannot be negative, i.e :math:`0 \\leq \\text{dCor} \\leq 1`.
Results have been tested against the
`energy <https://cran.r-project.org/web/packages/energy/energy.pdf>`_
R package.
References
----------
* https://en.wikipedia.org/wiki/Distance_correlation
* Székely, G. J., Rizzo, M. L., & Bakirov, N. K. (2007).
Measuring and testing dependence by correlation of distances.
The annals of statistics, 35(6), 2769-2794.
* https://gist.github.com/satra/aa3d19a12b74e9ab7941
* https://gist.github.com/wladston/c931b1495184fbb99bec
Examples
--------
1. With two 1D vectors
>>> from pingouin import distance_corr
>>> a = [1, 2, 3, 4, 5]
>>> b = [1, 2, 9, 4, 4]
>>> dcor, pval = distance_corr(a, b, seed=9)
>>> print(round(dcor, 3), pval)
0.763 0.312
2. With two 2D arrays and no p-value
>>> import numpy as np
>>> np.random.seed(123)
>>> from pingouin import distance_corr
>>> a = np.random.random((10, 10))
>>> b = np.random.random((10, 10))
>>> round(distance_corr(a, b, n_boot=None), 3)
0.88
"""
assert alternative in [
"two-sided",
"greater",
"less",
], "Alternative must be one of 'two-sided' (default), 'greater' or 'less'."
x = np.asarray(x)
y = np.asarray(y)
# Check for NaN values
if any([np.isnan(np.min(x)), np.isnan(np.min(y))]):
raise ValueError("Input arrays must not contain NaN values.")
if x.ndim == 1:
x = x[:, None]
if y.ndim == 1:
y = y[:, None]
assert x.shape[0] == y.shape[0], "x and y must have same number of samples"
# Extract number of samples
n = x.shape[0]
n2 = n**2
# Process first array to avoid redundancy when performing bootstrap
a = squareform(pdist(x, metric="euclidean"))
A = a - a.mean(axis=0)[None, :] - a.mean(axis=1)[:, None] + a.mean()
dcov2_xx = np.vdot(A, A) / n2
# Process second array and compute final distance correlation
dcor = _dcorr(y, n2, A, dcov2_xx)
# Compute one-sided p-value using a bootstrap procedure
if n_boot is not None and n_boot > 1:
# Define random seed and permutation
rng = np.random.RandomState(seed)
bootsam = rng.random_sample((n_boot, n)).argsort(axis=1)
bootstat = np.empty(n_boot)
for i in range(n_boot):
bootstat[i] = _dcorr(y[bootsam[i, :]], n2, A, dcov2_xx)
pval = _perm_pval(bootstat, dcor, alternative=alternative)
return dcor, pval
else:
return dcor
